The Cancellation Law for Groups

On the Basic Theorems Regarding Groups page we looked at a whole bunch of properties of groups. We will now look at another important property of groups called the cancellation law.

Theorem 1 (The Cancellation Law for Groups): Let $(G, \cdot)$ be a group and let $a, b, c \in G$. If $a \cdot b = a \cdot c$ or $b \cdot a = c \cdot a$ then $b = c$.

• Proof: Let $a^{-1} \in G$ denote the inverse of $a$ under $\cdot$. Suppose that $a \cdot b = a \cdot c$. Then:

(1)

\begin{align} \quad a \cdot b &= a \cdot c \\ \quad (a^{-1} \cdot a) \cdot b &= (a^{-1} \cdot a) \cdot c \\ \quad e \cdot b &= e \cdot c \\ \quad b &= c \end{align}

• Similarly, suppose now that $b \cdot a = c \cdot a$. Then:

(2)

\begin{align} \quad b \cdot a &= c \cdot a \\ \quad (b \cdot a) \cdot a^{-1} &= (c \cdot a) \cdot a^{-1} \\ \quad b \cdot (a \cdot a^{-1}) &= c \cdot (a \cdot a^{-1}) \\ \quad b \cdot e &= c \cdot e \\ \quad b &= c \quad \blacksquare \end{align}

It is very important to note that the cancellation law holds with regards to the operation $\cdot$ for any group $(G, \cdot)$. We will see that the cancellation law does not necessarily hold with respect to an operation on a set when we look at algebraic structures with two defined operations.

It is also important to note that if $a \cdot b = c \cdot a$ or $b \cdot a = a \cdot c$ then we cannot necessarily deduce that $b = c$ because we would then require the additional property that $\cdot$ is commutative which is not one of the group axioms (but instead one of the Abelian group axioms).